# PrimeOmega

PrimeOmega[n]

gives the number of prime factors counting multiplicities in n.

# Details and Options

• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• PrimeOmega gives the number of prime factors of an integer with multiplicity.
• For a number with a unit and primes, PrimeOmega[n] returns k1++km.
• With the setting , PrimeOmega gives the number of Gaussian prime factors.
• PrimeOmega[m+In] automatically works over Gaussian integers.

# Examples

open allclose all

## Basic Examples(2)

Compute PrimeOmega at 30:

Plot the PrimeOmega sequence for the first 100 numbers:

## Scope(8)

### Numerical Evaluation(4)

PrimeOmega works over integers:

Gaussian integers:

Compute for large integers:

### Symbolic Manipulation(4)

Reduce expressions:

Solve equations:

Identify the PrimeOmega sequence:

## Options(1)

### GaussianIntegers(1)

Compute PrimeOmega over integers:

Gaussian integers:

## Applications(6)

### Basic Applications(2)

Table of the values of PrimeOmega for the integers up to 100:

Histogram of the values of PrimeOmega:

### Number Theory(4)

Use PrimeOmega to test for a prime number:

Use PrimeOmega to compute LiouvilleLambda:

Compare with:

Plot the average over values of PrimeOmega for different ranges of integer arguments:

The Fourier statistics of the PrimeOmega sequence:

## Properties & Relations(5)

Use FactorInteger to find the number of prime factors counting multiplicities:

PrimeOmega is a completely additive function:

PrimeOmega gives the exponent for a prime power:

PrimeOmega and PrimeNu are equivalent when the argument is square-free:

PrimeOmega is always greater than or equal to PrimeNu:

## Possible Issues(1)

PrimeOmega is not defined at 0:

## Neat Examples(2)

Plot the arguments of the Fourier transform of PrimeOmega:

Plot the Ulam Spiral of PrimeOmega:

Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

#### Text

Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

#### CMS

Wolfram Language. 2008. "PrimeOmega." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimeOmega.html.

#### APA

Wolfram Language. (2008). PrimeOmega. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimeOmega.html

#### BibTeX

@misc{reference.wolfram_2024_primeomega, author="Wolfram Research", title="{PrimeOmega}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeOmega.html}", note=[Accessed: 12-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_primeomega, organization={Wolfram Research}, title={PrimeOmega}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeOmega.html}, note=[Accessed: 12-July-2024 ]}